Two-sided a posteriori error bounds for electro-magnetostatic problems. (English) Zbl 1288.35461
J. Math. Sci., New York 166, No. 1, 53-62 (2010) and Zap. Nauchn. Semin. POMI 370, 94-110 (2009).
Summary: This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between exact and approximate solutions of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not invoke specific properties of the approximation method. For this reason, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for elliptic problems.
MSC:
| 35Q74 | PDEs in connection with mechanics of deformable solids |
References:
| [1] | A. Alonso and A. Valli, ”Some remarks on the characterization of the space of tangential traces of H (rot, ) and the construction of an extension operator,” Manuscripta Math., 89, 159-178 (1996). · Zbl 0856.46019 · doi:10.1007/BF02567511 |
| [2] | I. Anjam, O. Mali, A. Muzalevsky, P. Neittaanmäki, and S. Repin, ”A posteriori error estimates for a Maxwell type problem,” Russ. J. Numer. Anal. Math. Modeling, 24 (5), 395-408 (2009). · Zbl 1196.78015 |
| [3] | A. Buffa and J. P. Ciarlet, ”On traces for functional spaces related to Maxwell’s equations. Part I: an integrations by parts formula in Lipschitz polyhedra,” Math Methods Appl. Sci., 24, 9-30 (2001). · Zbl 0998.46012 · doi:10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2 |
| [4] | A. Buffa, M. Costabel, and D. Sheen, ”On traces for H (curl, ) in Lipschitz domains,” J. Math. Anal. Appl., 276, 845-867 (2002). · Zbl 1106.35304 · doi:10.1016/S0022-247X(02)00455-9 |
| [5] | P. Kuhn, Die Maxwellgleichung mit wechselnden Randbedingungen, Shaker Verlag, Essen (1999). |
| [6] | P. Kuhn and D. Pauly, ”Generalized Maxwell equations in exterior domains. I: regularity results, trace theorems, and static solution theory,” Reports of the Department of Mathematical Information Technology, University of Jyväskylä, Finland (2007). |
| [7] | P. Kuhn and D. Pauly, ”Regularity results for generalized electro-magnetic problems,” Analysis (Munich) (2009) (accepted for publication). · Zbl 1225.35229 |
| [8] | R. Leis, Initial Boundary Value Problems in Mathematical Physics, Teubner, Stuttgart (1986). · Zbl 0599.35001 |
| [9] | A. Milani and R. Picard, ”Decomposition theorems and their apllications to nonlinear electro- and magnetostatic boundary value problems,” Lect. Notes Math., 1357, 317-340 (1988). · doi:10.1007/BFb0082873 |
| [10] | D. Pauly, ”Generalized electro-magnetostatic in nonsmooth exterior domains,” Analysis (Munich), 27 (4), 425-464 (2007) · Zbl 1132.35487 |
| [11] | D. Pauly, ”Hodge-Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media,” Math. Methods Appl. Sci., 31, 1509-1543 (2008). · Zbl 1159.58002 · doi:10.1002/mma.982 |
| [12] | D. Pauly and S. Repin, ”Functional a posteriori error estimates for elliptic problems in exterior domains,” J. Math. Sci. (N. Y.), 162 (3), 393-406 (2009). · doi:10.1007/s10958-009-9643-4 |
| [13] | D. Pauly and T. Rossi, ”Computation of generalized time-periodic waves using differential forms, controllability, least-squares formulation, conjugate gradient method and discrete exterior calculus: part I - theoretical considerations,” Reports of the Department of Mathematical Information Technology, University of Jyväskylä, Finland (2008). |
| [14] | R. Picard, ”Randwertaufgaben der verallgemeinerten Potentialtheorie,” Math. Meth. Apply. Sci., 3, 218-228 (1981). · Zbl 0466.31016 · doi:10.1002/mma.1670030116 |
| [15] | R. Picard, ”On the boundary value problems of electro- and magnetostatics,” Proc. Roy. Soc. Edinburgh Sect. A, 92, 165-174 (1982). · Zbl 0516.35023 · doi:10.1017/S0308210500020023 |
| [16] | R. Picard, ”An elementary proof for a compact imbedding result in generalized electromagnetic theory,” Math. Z., 187, 151-164 (1984). · Zbl 0527.58038 · doi:10.1007/BF01161700 |
| [17] | R. Picard, ”Some decomposition theorems and their applications to nonlinear potential theory and Hodge theory,” Math. Methods Appl. Sci., 12, 35-53 (1990). · Zbl 0702.35212 · doi:10.1002/mma.1670120103 |
| [18] | R. Picard, N. Weck, and K. J. Witsch, ”Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles,” Analysis (Munich), 21, 231-263 (2001). · Zbl 1075.35085 |
| [19] | S. Repin, ”A posteriori error estimates for variational problems with uniformly convex functionals,” Math. Comp., 69 (230), 481-500 (2000). · Zbl 0949.65070 · doi:10.1090/S0025-5718-99-01190-4 |
| [20] | S. Repin, A Posteriori Estimates for Partial Differential Equations, Walter de Gruyter, Berlin (2008). · Zbl 1162.65001 |
| [21] | C. Weber, ”A local compactness theorem for Maxwell’s equations,” Math. Methods Appl. Sci., 2, 12-25 (1980). · Zbl 0432.35032 · doi:10.1002/mma.1670020103 |
| [22] | N. Weck, ”Maxwell’s boundary value problems on Riemannian manifolds with nonsmooth boundaries,” J. Math. Anal. Appl., 46, 410-437 (1974). · Zbl 0281.35022 · doi:10.1016/0022-247X(74)90250-9 |
| [23] | N. Weck, ”Traces of differential forms on Lipschitz boundaries,” Analysis (Munich), 24, 147-169 (2004). · Zbl 1187.35023 |
| [24] | K. J. Witsch, ”A remark on a compactness result in electromagnetic theory,” Math. Methods Appl. Sci., 16, 123-129 (1993). · Zbl 0778.35105 · doi:10.1002/mma.1670160205 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
